At 20, the only firearms you can lawfully buy will be long guns - rifles and shotguns. You need to have a state-issued photo ID (driver's license), and know your social security number. You can't be convicted of any violent crimes or felonies, can't be subject to a restraining order, and can't have been adjudicated insane or incompetent by a court.
You will be subject to a background check, and potentially a waiting period, depending on the state. As for what you "should" buy, that depends on your purpose. For hunting game, a .270 rifle is a good all-round shooter, but .308 is another great caliber.
For bird hunting (dove, pheasant, quail, duck), a 12- or 20-gauge shotgun would be good. Pumps are cheap, but a good semi-auto is more useful. Regardless of what you buy, take a safety class and spend a lot of time at a range practicing.
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